for \(E_t : y^2 = x^3 + tx + 1\), let
\[ f(x,t) = \int_{(0,1)}^{(x,z)} \frac{\mathrm{d}x}{y} \]
Here we are integrating over two constant sections \((0,1),(x,z)\).
We have a second order differential equation satisfied fy \(f(x,t)\)
\[\begin{align*} & \frac{-243 t + 108 t^3 - 36 t^4}{-19683 - 8748 t^3 - 1296 t^6 - 64 t^9} D_t^0 f + \frac{-36 t}{729 + 216 t^3 + 16 t^6} D_t^1 f + \frac{-9}{27 + 4 t^3} D_t^2 f \\ = & \frac{1458 - 648 t^2 + 216 t^3}{19683 + 8748 t^3 + 1296 t^6 + 64 t^9} z^{-1} + \frac{26244 t - 11664 t^3 + 3888 t^4}{-531441 - 314928 t^3 - 69984 t^6 - 6912 t^9 - 256 t^{12}} x^1 z^{-1} \\ + & \frac{-1458 + 648 t^2 - 216 t^3}{19683 + 8748 t^3 + 1296 t^6 + 64 t^9} z^0 + \frac{-3888 t^4 + 1728 t^6 - 576 t^7}{531441 + 314928 t^3 + 69984 t^6 + 6912 t^9 + 256 t^{12}} x^1 z^1\\ + & \frac{26244 t^2 - 11664 t^4 + 3888 t^5}{531441 + 314928 t^3 + 69984 t^6 + 6912 t^9 + 256 t^{12}} x^2 z^1 + \frac{-1458 + 648 t^2 - 216 t^3}{19683 + 8748 t^3 + 1296 t^6 + 64 t^9} x^3 z^1 \\ + & \frac{-26244 t^2 + 11664 t^4 - 3888 t^5}{-531441 t - 314928 t^4 - 69984 t^7 - 6912 t^{10} - 256 t^{13}} x^4 z^1 \end{align*}\]
here both sections are assumed to be constant sections. If not it should involve higher derivatives of \(x\) and \(z\).
Proposition The left side of this DE only depends on \(t\). Can be proved by observing matrices.
I suddenly realize it makes no sense to take partial derivative of \(f(x,t)\), since the coordinate \(x\) has to be a function of \(t\). What we should do is to take two sections \(x_1(t)\) and \(x_2(t)\), consider
\[ f_{x_1,x_2}(t) = \int_{x_1(t)}^{x_2(t)} \omega_t. \]
Let’s review the theory, from a family of curves \(\pi: X\to S\) we have the universal extension representing all integrals
\[0\to \pi^*\mathcal{V}\to \mathcal{E}\to \mathcal{O}_X\to 0\]
where \(\mathcal{V}= \left(R^1\pi_*\Omega_{X/S}^\bullet\right)^\vee\in \mathrm{Conn}(S/K)\). We have two sections \(x_1, x_2 : S\to X\) and we require \(x_1^*\mathcal{E}\) splits. This along with the connection on \(\mathcal{E}\) being flat uniquely determines \(\mathcal{E}\).
\[\Lambda_\mathcal{E}= \begin{pmatrix} \Theta_{\mathcal{V}}\,\mathrm{d}{t} & \underline{g}\,\mathrm{d}{t}+\underline{h}\,\mathrm{d}{x} \\ & \\ \end{pmatrix} \in \mathrm{Hom}_{\mathcal{O}_X}\left(\mathcal{O}_X^{n+1}, \Omega_{X/K}^{n+1}\right). \]
Flatness equation of \(\mathcal{E}\) expands to
\[\begin{align*} \Theta_x &= 0 \\ \underline{g}_x &= \underline{h}_t + \Theta \underline{h} \end{align*}\]
which together with the definition of a Gauss-Manin connection data \((G,h,g)\)
\[\underline{h}_t = G^T\underline{h} + \underline{g}_x\]
implies that
Theorem. \(\mathcal{V}\) being the dual of Gauss-Manin connection in the sense that \(\left(-\Theta^T,h,g\right)\) is a Gauss-Manin data, is equivalent to the connection \(\mathcal{E}\) being flat.
Being Gauss-Manin connection does not determine the connection \(\mathcal{E}\). If you think it as families of integration, you need to specify the base points. We can do so by requiring that the pullback \(s_1^*\mathcal{E}\) splits.
Note that there can be a lot of coordinates \(x^{(i)}\) in \(X_t\), but we are assuming the one-dimensional situation that they all reduces to some single differential \(\,\mathrm{d}{x^{(0)}}\) (which is the \(\,\mathrm{d}{x}\) mentioned above).
\[\,\mathrm{d}{x^{(i)}} = \phi_i(x) \,\mathrm{d}{x^{(0)}}.\]
The equation for the splitting of the section \(s_1(t) = \left(s_1^{(0)}(t), s_1^{(1)}(t), \ldots, s_1^{(n)}(t)\right)\) is
\[s_1^*g + s_1^*\omega = g(s_1(t),t)\,\mathrm{d}{t} + h(s_1(t),t)\,\mathrm{d}{s_1^{(0)}} = 0\]
which reduces to
\[g(s_1(t),t) + h(s_1(t),t)\frac{\partial s_1^{(0)}}{\partial t} = 0.\]
We denote this equation by (SP), meaning splitting.
Now in order to choose \(g\) that satisfy both (GM) and (SP), first choose arbitrary \(g_0\) that satisfy (GM), i.e. \((-\Theta^T, h, g_0)\) is a GM data. Compute \(r(t)\) as
\[ r(t) = g_0(s_1(t),t) + h(s_1(t),t)\frac{\partial s_1^{(0)}}{\partial t}.\]
then define \(g(x,t) = g_0(x,t) - r(t)\). This gives a GM data \((-\Theta^T, h, g)\) that satisfies both (GM) and (SP).
Theorem. For any section \(s_1:S\to X\), we can find a Gauss-Manin connection data \((G,h,g)\) that satisfies both (GM) and (SP), i.e.
\(h_t = G^T h + g_x\),
\(g(s_1(t),t) + h(s_1(t),t)\frac{\partial s_1^{(0)}}{\partial t} = 0\) for all \(t\in S\).
The connection \(\mathcal{E}\) associated to this GM data is called based at \(s_1\) and denoted by \(\mathcal{E}_{s_1}\).
Given two sections \(s_1,s_2:S\to X\), choose \(\mathcal{E}_{s_1}\) with data \((-\Theta^T, h, g)\) as above. Then the function
\[ f(t) = \int_{s_1(t)}^{s_2(t)} \underline{\omega}_t \]
satisfies the DE
\[\frac{d}{dt} \begin{pmatrix} f \\ 1 \end{pmatrix} = \begin{pmatrix} \Theta & g_{s_1}(s_2(t),t) + h(s_2(t),t)\frac{\partial s_2^{(0)}}{\partial t} \\ 0 & 0 \end{pmatrix} \begin{pmatrix} f \\ 1 \end{pmatrix} \]
whose matrix we denote by \(\Lambda_1\), using convention \(\Lambda_0 = I_{n+1}\). Inductively we can define
\[ \Lambda_{n+1} = \frac{d}{dt}(\Lambda_n) + \Lambda_n\Lambda_1. \]
Lemma.(Getting DE from differential system)
For any differential system \(D_t f = \Lambda_1 f\), where \(\Lambda_1\in M_n(R)\) in certain ring \(R\), we can get a degree \(n\) homogeneous differential equation for \(f_i\) as
\[\sum_{k=0}^n (-1)^k \det \begin{pmatrix} (\Lambda_n)_i \\ \vdots \; \text{skip }\Lambda_k \\ (\Lambda_0)_i \\ \end{pmatrix} f_i^{(k)} = 0 \]
For any non-homogeneous differential system of the form
\[D_t \begin{pmatrix} f \\ 1 \end{pmatrix} = \begin{pmatrix} A & b \\ 0 & 0 \end{pmatrix} \begin{pmatrix} f \\ 1 \end{pmatrix} \]
where \(\Lambda_1 = \begin{pmatrix} A & b \\ 0 & 0 \end{pmatrix}\), we can get a degree \(n\) non-homogeneous differential equation for \(f_i\) as
\[\sum_{k=0}^n (-1)^k \det \begin{pmatrix} & (\Lambda_n)_i & \\ \vdots & \text{skip }\Lambda_k & \vdots \\ & (\Lambda_0)_i & \\ & & 1 \end{pmatrix} f_i^{(k)} = (-1)^{n} \det \begin{pmatrix} (\Lambda_n)_i \\ \dots \\ (\Lambda_0)_i \\ \end{pmatrix} \]
Theorem. The differential equations for \(f(t) = \int_{s_1(t)}^{s_2(t)} \underline{\omega}_t\) is given by the following non-homogeneous differential system
\[D_t \begin{pmatrix} f \\ 1 \end{pmatrix} = \begin{pmatrix} \Theta(t) & u_{s_1,s_2}(t) \\ 0 & 0 \end{pmatrix} \begin{pmatrix} f \\ 1 \end{pmatrix} \]
where we pick any \(g\) that satisfies (GM) and define the function
\[u_{s_1,s_2}(t) = g(s_2,t) - g(s_1,t) + h(s_2,t)\frac{\partial s_2^{(0)}}{\partial t} - h(s_1,t)\frac{\partial s_1^{(0)}}{\partial t}.\]